Abstract
This work recollects a non-universal set of quantum gates described by higher-dimensional Spin groups. They are also directly related with matchgates in theory of quantum computations and complexity. Various processes of quantum state distribution along a chain such as perfect state transfer and different types of quantum walks can be effectively modeled on classical computer using such approach.
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Valiant, L.G.: Quantum computers that can be simulated classically in polynomial time. In: Proceedings of 33rd Annual ACM STOC, pp. 114–123 (2001)
Terhal, B.M., DiVincenzo, D.P.: Classical simulation of noninteracting-fermion quantum circuits. Phys. Rev. A 65, 032325 (2002)
Knill, E.: Fermionic linear optics and matchgates (2001). Preprint arXiv:quant-ph/0108033
Kitaev, A.Y.: Unpaired Majorana fermions in quantum wires. Phys.-Usp. (Suppl.) 44, 131–136 (2001)
Bravyi, S., Kitaev, A.: Fermionic quantum computation. Ann. Phys. 298, 210–226 (2002)
Vlasov, A.Y.: Clifford algebras and universal sets of quantum gates. Phys. Rev. A 63, 054302 (2001)
Jozsa, R., Miyake, A.: Matchgates and classical simulation of quantum circuits. Proc. R. Soc. Lond. Ser. A 464, 3089–3106 (2008)
Jozsa, R., Kraus, B., Miyake, A., Watrous, J.: Matchgate and space-bounded quantum computations are equivalent. Proc. R. Soc. Lond. Ser. A 466, 809–830 (2010)
Jozsa, R., Miyake, A., Strelchuk, S.: Jordan–Wigner formalism for arbitrary 2-input 2-output matchgates and their classical simulation. Quantum Inf. Comput. 15, 541–556 (2015)
Brod, D.J.: Efficient classical simulation of matchgate circuits with generalized inputs and measurements. Phys. Rev. A 93, 062332 (2016)
Vlasov, A.Y.: Quantum gates and Clifford algebras (1999). Preprint arXiv:quant-ph/9907079
Wilczek, F.: Majorana returns. Nat. Phys. 5, 614–618 (2009)
Nayak, C., Simon, S., Stern, A., Freedman, M., Das, Sarma S.: Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008)
Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)
Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proceedings of 33rd Annual ACM STOC, pp. 60–69 (2001)
Kempe, J.: Quantum random walks—an introductory overview. Contemp. Phys. 44, 307–327 (2003)
Kendon, V.: Decoherence in quantum walks—a review. Math. Struct. Comput. Sci. 17(6), 1169–1220 (2006)
Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11(5), 1015–1106 (2012)
Barenco, A., Deutsch, D., Ekert, A.K., Jozsa, R.: Conditional quantum dynamics and logic gates. Phys. Rev. Lett. 74, 4083–4086 (1995)
Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: Proceedings of 45th Annual IEEE FOCS, pp. 32–41 (2004)
Portugal, R., Santos, R.A.M., Fernandes, T.D., Gonçalves, D.N.: The staggered quantum walk model. Quantum Inf. Process. 15(1), 85–101 (2016)
Portugal, R.: Establishing the equivalence between Szegedy’s and coined quantum walks using the staggered model. Quantum Inf. Process. 15(4), 1387–1409 (2016)
Wong, T.: Equivalence of Szegedy’s and coined quantum walks. Quantum Inf. Process. 16(9), 215 (2017)
Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)
Bose, S.: Quantum communication through an unmodulated spin chain. Phys. Rev. Lett. 91, 207901 (2003)
Christandl, M., Datta, N., Ekert, A., Landahl, A.J.: Perfect state transfer in quantum spin networks. Phys. Rev. Lett. 92, 187902 (2004)
Christandl, M., Datta, N., Dorlas, T.C., Ekert, A., Kay, A., Landahl, A.J.: Perfect transfer of arbitrary states in quantum spin networks. Phys. Rev. A 71, 032312 (2005)
Burgarth, D., Bose, S.: Conclusive and arbitrarily perfect quantum state transfer using parallel spin chain channels. Phys. Rev. A 71, 052315 (2005)
Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics. III. Quantum Mechanics. Addison-Wesley, Reading (1965)
Zimborás, Z., Faccin, M., Kádár, Z., Whitfield, J., Lanyon, B., Biamonte, J.: Quantum transport enhancement by time-reversal symmetry breaking. Sci. Rep. 3, 02361 (2013)
Lu, D., Biamonte, J.D., Li, J., Li, H., Johnson, T.H., Bergholm, V., Faccin, M., Zimborás, Z., Laflamme, R., Baugh, J., Lloyd, S.: Chiral quantum walks. Phys. Rev. A 93, 042302 (2016)
Stancil, D.D., Prabhakar, A.: Spin Waves: Theory and Applications. Springer, New York (2009)
Thompson, K.F., Gokler, C., Lloyd, S., Shor, P.W.: Time independent universal computing with spin chains: quantum plinko machine. New J. Phys. 18, 073044 (2016)
Kontorovich, V.M., Tsukernik, V.M.: Spiral structure in one-dimensional chain of spins. Sov. Phys. JETP 25, 960–964 (1967)
Derzhko, O., Verkholyak, T., Krokhmalskii, T., Büttner, H.: Dynamic probes of quantum spin chains with the Dzyaloshinskii–Moriya interaction. Phys. Rev. B 73, 214407 (2006)
Weyl, H.: The Theory of Groups and Quantum Mechanics. Dover Publications, New York (1931)
Schumacher, B., Werner, R.F.: Reversible quantum cellular automata (2004). Preprint arxiv:quant-ph/0405174
Jordan, P., Wigner, E.: Über das Paulische Äquivalenzverbot. Zeitschrift für Physik 47, 631–651 (1928)
Lang, S.: Algebra. Springer, New York (2002)
Gilbert, J.E., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)
DiVincenzo, D.P.: Two-bit gates are universal for quantum computation. Phys. Rev. A 51, 1015–1022 (1995)
Postnikov, M.M.: Lie Groups and Lie Algebras. Mir Publishers, Moscow (1986)
Vlasov, A.Y.: Permanents, bosons and linear optics. Laser Phys. Lett. 14, 103001 (2017)
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The author is grateful to anonymous referees for corrections and suggestions to initial version of the article.
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Vlasov, A.Y. Effective simulation of state distribution in qubit chains. Quantum Inf Process 17, 269 (2018). https://doi.org/10.1007/s11128-018-2036-1
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DOI: https://doi.org/10.1007/s11128-018-2036-1